Introduction

With the rapid development of information technology and increasing demand for the miniaturization of devices, two-dimensional (2D) materials with low power consumption, high storage density, and integration have become the focus of intensive research1,2,3,4. Among them, 2D multiferroics combine the advantages of multiple ferroic orders and hold promise in multi-stage storage and nonvolatile logic devices5,6,7. They are a special class of materials that simultaneously exhibit two or more primary ferroic orders, such as ferroelectricity, ferromagnetism, ferroelasticity, and ferrovalley, and there exists coupling between them, which can be mutually controlled. In particular, the coupling between ferroelectric (FE) and ferromagnetic (FM) orders leads to the magnetoelectric (ME) coupling effect, which offers a unique opportunity to control the spin by applying an electric field. This kind of ME coupling could be used to develop new-generation memory devices in which data can be written electrically and read magnetically, avoiding the drawback of high writing energy in conventional magnetic random-access memory8,9,10.

Depending on the microscopic FE mechanism, multiferroic materials can be divided into two categories, i.e., type-I and type-II. In type-I multiferroics, the ferroelectricity and magnetism originate from different atoms. The coupling between electric dipoles and magnetic spins is mediated by spin-lattice interaction, usually leading to weak ME coupling, but often exhibits considerable spontaneous polarization and high transition temperature. To date, several type-I multiferroics such as CrN11, MIMIIP2X6 (MI, MII = metal elements, X = O/S/Se/Te)12, and ReWCl613 have been predicted theoretically, while multiferroic has been confirmed experimentally in 2D CuCrP2S614. In type-II multiferroics, FE order generally arises from special magnetic order, and the direct correlation between polarization and magnetization promises strong ME coupling. Generally, special magnetic order, such as spiral or complex collinear magnetic order, could lead to spin and charge frustration, which significantly reduces the intensity of FE polarization and transition temperature. Hf2VC2F215, MnRe2O816, and VX2 (X = Cl/Br/I)17 monolayers are predicted to exhibit 120° non-collinear Y-type antiferromagnetic (AFM) spin spiral orders, which can induce FE polarization perpendicular to the spin spiral plane. The polarization switching can realize the reversal of the spin spiral chirality in MnRe2O8 and VX2 monolayers, demonstrating strong magnetoelectric coupling behavior. In addition, Bao et al. theoretically demonstrated that synthesized FeOCl monolayer has a cycloidal spin spiral with a propagation vector of (0.274, 0.5, 0) located in the ac plane, and the induced out-of-plane ferroelectricity can switch the spin spiral chirality through an electric field18. There was also experimental evidence that the NiI2 monolayer exhibited proper screw state with a given handedness coupled to the charge degrees of freedom, producing a chirality-controlled electric polarization with a transition temperature of 21 K19. Since both type-I and type-II multiferroics have advantages and shortcomings, continuous search for single-phase 2D multiferroic materials with excellent FE–FM multiferroic properties, i.e., strong ME coupling, high Curie temperature, and FE polarization intensity, is still a challenge for this fast-growing field.

Intercalation is a chemical process in which foreign species insert into crystal gaps to eventually form hybrid intercalation compounds. In 2D layered materials with van der Waals (vdW) gap, intercalation reaction has been proven as a powerful approach for synthesizing single-phase crystals with the coexistence of multiple order parameters20. A family of 2D MA2Z4 materials was obtained by intercalating a MoS2-type MZ2 monolayer into InSe-type A2Z2 bilayers. Among them, 2D MnBi2Te4 is an AFM axion/Chern insulator with quantized anomalous Hall conductivity21,22,23, while PdBi2Te4 is a superconducting topological metal24. Nontrivial topological properties, FM semiconductors, Ising superconductivity, and robust electron valleys are also found in SrGa2Te4, NbSi2N4, TiSi2N4, MoSi2P4, MoSi2As4, WSi2P4, and WSi2As421,25. In addition, the Kagome lattice formed by Ta8Se12 (66.7% Ta-intercalated) stabilizes the charge-density-wave and exhibits FM metallicity20, while Cr-intercalated 2H-TaS2 (Cr1/3TaS2) is a chiral crystal, which has a helical spin texture and continuously transforms into a chiral soliton lattice, eventually reaches a forced FM state under an external magnetic field26. In addition, 2D AgCrS2 composed of Ag intercalated CrS2 bilayers has been experimentally demonstrated to be stable and exhibit superionic and FM behavior at room temperature27,28. Briefly speaking, intercalation breaks the inversion symmetry of the system, changes the bonding order from vdW to ionic or covalent type, and modulates the band structures of the host lattice and intercalated species.

Inspired by the exotic properties triggered by unique intercalated architectures, herein we propose a general approach to systematically design 2D multiferroic materials by non-centrosymmetric intercalation of 3d/4d transition metal ions (A) into the 2H/1T phase of TMD bilayers (MX2). By filtering a total of 960 possible combinations, we have reproduced the previously reported CoMo2S429, AgCr2S4, and AgCr2Se428, and further unveiled 40 dynamically and thermodynamically stable 2D AM2X4 ferroelectrics with low FE switching barrier (<200 meV/f.u.). Among them, different magnetic orders such as FM, AFM, and ferrimagnetic (FiM) are identified in the 21 FE systems. Depending on the position of the magnetic ions (intercalated A ions or host M atoms), we classify these 21 multiferroics into three types and investigate their different ME coupling behavior. Our intercalation strategy opens an avenue to develop novel 2D materials with multiferroic properties and explores other fascinating topological phenomena.

Results and discussions

Screening of 2D non-centrosymmetric intercalated compounds

Transition metal disulfide compounds (TMDs) are a series of layered crystals with a chemical formula MX2 (M = transition metal, X = S/Se/Te), in which the neighboring MX2 layers interact weakly with each other via vdW interaction. Such weak interlayer interaction means that different thicknesses of ultrathin TMD films can be obtained in experiments using exfoliation approaches. In general, MX2 monolayers crystallize into three main phases: 1T (triangular) phase with D3d symmetry, 2H (hexagonal) phase with D3h symmetry, and distorted 1T phase (1T’ phase). Some MX2 with 1T and 2H phases exhibit tunable band gaps, high carrier mobility, and strong intrinsic spin-orbit coupling, making them attractive candidates in electronic, optical, and spintronic devices30. Owing to the existence of the vdW gap, the MX2 compounds can be easily intercalated by atoms, molecules, or atomic layers, and their properties can also be modulated via intercalation. For example, previous studies have shown that after the intercalation of Ag ions into a 2H–CrS2 bilayer or of Cu into a 2H-CrX2 (X = S/Se) bilayer, the intercalation compounds AgCrX2 and CuCrX2 transits to multiferroic state due to the breaking of inversion symmetry by Ag/Cu ions28,31; 2D CoMo2S4 also shows multiferroic behavior as the Co ions are intercalated into the 1T-MoS2 bilayer and the two FE ground states are energetically degenerate with opposite chirality of the Dzyaloshinskii Moriya interaction (DMI)29.

Inspired by these results, we have constructed a series of hybrid intercalation compounds using 2H or 1T phase MX2 bilayers with AB stacking as host lattice and A atoms as guest species. When the intercalated A atoms uniformly occupy the tetragonal-like vacancies in the interlayer space of bilayer MX2, an A atom coordinates with one X atom from the top layer and three X atoms from the bottom layer. This leads to an unequal distance between A and the two MX2 layers, thereby breaking the central inversion symmetry of the system and generating spontaneous polarization. The tetrahedral crystal field can be switched by a 180° rotation when an A ion bonds with three X atoms in the top layer and one X atom in the bottom layer, leading to polarization reversal. The selection of the MX2 layer and A atoms is summarized in Supplementary Table 1, and a total of 960 non-centrosymmetric intercalation compounds were obtained with all combinations of the MX2 layer and A atom. The representative crystal structures of 2H phase compounds AM2X4 (H-AM2X4) and 1T phase compound AM2X4 (T-AM2X4), as well as the corresponding FE switching mechanism, are shown in Fig. 1.

Fig. 1: The structural diagram of T-AM2X4 and H-AM2X4.
figure 1

a T-AM2X4 and b H-AM2X4 structure diagrams obtained by intercalating A atom into 1T- and 2H-MX2 bilayers. The A and A′ represent the atomic positions of the intercalated atoms in FE1 and FE2 states, respectively, and the blue arrows describe the relative displacement of the intercalated atoms between FE1 and FE2 states.

Figure 2 schematically illustrates the procedure of high-throughput DFT calculations on the structures and ferroic properties of 2D 2H/1T-AM2X4 systems. Firstly, we performed a full structural optimization to confirm that the intercalated A ions in all 960 AM2X4 compounds deviated from the center position and caused spontaneous polarization. Secondly, we evaluated the dynamic and thermodynamic stabilities of these AM2X4 compounds by computing their phonon dispersion and formation energy. As can be seen in Supplementary Fig. 1, there is no negative frequency throughout the entire Brillouin zone of 104 2D AM2X4 compounds. Among them, the formation energies of 100 systems are negative (Supplementary Table 2), indicating that they are dynamically and thermodynamically stable at ambient conditions. Thirdly, the smaller the FE switching barrier, the less energy is consumed for polarization reversal. Hence, we set a convergence criterion for the polarization reversal barrier of 200 meV/f.u., which is equivalent to ~kBT per atom at ambient conditions. It is noteworthy that we have reproduced the previously reported T-AgCr2S4, T-AgCr2Se4, T-CuCr2S4, T-CuCr2Se4, and H-CoMo2S4. Thus, these systems will not be further discussed in the rest of this paper. Fourthly, for the remaining 40 FE candidates after step-three screening, we considered two possible antiferroelectric (AFE) states in a 2 × 2 × 1 supercell to determine their FE ground states (Supplementary Fig. 2 and Supplementary Table 3). The FE states were more stable than the AFE states for most compounds, except that T-CoTi2Te4 prefers an AFE ground state. However, the AFE and FE phases of T-CoTi2Te4 are separated by a switching energy barrier of about 219.44 meV/f.u. (Supplementary Fig. 3), which is high enough to prevent a spontaneous FE-to-AFE transition at room temperature. Therefore, the FE phase of T-CoTi2Te4 can still exist as a metastable state. In conclusion, we screen out 40 previously unknown members of stable 2D FE materials. Their electronic, magnetic, and ME coupling properties will be investigated further.

Fig. 2: Schematic flowchart of the high-throughput first-principles calculations to screen the ferroic materials among the intercalation compounds AM2X4.
figure 2

The FE, FM, AFM, and FiM represent ferroelectric, ferromagnetic, antiferromagnetic and ferrimagnetic, respectively.

The electronic band structures of these 40 candidate FE materials at the GGA + U level are displayed in Supplementary Fig. 4. Among them, the intercalation compounds of T-CdSc2Se4, T-CdRh2S4, T-CoSc2S4, and T-CoY2S4 are semiconductors with band gaps of 0.35, 0.14, 0.20, and 0.04 eV, respectively. In T-CdSc2Se4 and T-CdRh2S4, both the valence band maximum (VBM) and the conduction band minimum (CBM) are dominated by the ScSe2 and RhS2 layers, while the VBMs of T-CoSc2S4 and T-CoY2S4 are mainly contributed by the S atoms and the CBMs stem mainly from the intercalated Co atoms (Supplementary Fig. 6). Apart from the four semiconductors mentioned above, other intercalation compounds exhibit metallic or half-metallic behavior, since the 3d/4d orbitals of the intercalated transition metal ions are usually partially occupied and exhibit strong itinerant characteristics. Based on this finding, we will discuss the intrinsic FE behavior of semiconductors and metals/half-metals separately in the following content.

The intrinsic FE behaviors of AM2X4 monolayers

In-plane polarizations (Pin) along the [110] direction and out-of-plane polarizations (Pout) are observed in the semiconducting T-CdSc2Se4, T-CdRh2S4, T-CoSc2S4, and T-CoY2S4. The Pin of T-CdSc2Se4, T-CdRh2S4, T-CoSc2S4, and T-CoY2S4 calculated with the Berry phase method are 269.28 (2.30), 208.55 (1.36), 306.45 (2.16), and 128.26 (1.13) pC/m (eÅ), respectively, while the Pout evaluated by the dipole correction approach are 15.22 (0.13), 9.32 (0.06), 11.35 (0.08), and 13.62 (0.12) pC/m (eÅ). These values are comparable to the typical 2D FE semiconductor In2Se3 [Pin = 259.08 (2.36) pC/m (eÅ), Pout = 12.08 (0.11) pC/m (eÅ)]32. Similar to In2Se3, the interesting intercorrelated in-plane and out-of-plane ferroelectricity may also occur in these 2D FE semiconductors, where the out-of-plane electric field used to reverse the Pout also reverses the Pin, which has great potential for advanced memory devices and multi-directional controlled field-effect transistors33,34,35.

For metals or half-metals, there is only a Pout as the Pin is annihilated by free electrons. Using metallic T-PdZr2Se4 as an example, we discuss the mechanism of coexistence of ferroelectricity and metallicity. The metallic behavior of T-PdZr2Se4 was confirmed by calculating the electronic band structure, as displayed in Fig. 3a, b. The conduction electrons \({\rho }_{c}(\text{z})\) calculated by Eq. (3) are mainly distributed in the top and bottom ZrSe2 layers (Fig. 3c), which is consistent with the projected density of states (PDOS) results (Fig. 3b). However, the polarization electrons \({\rho }_{P}(\text{z})\) obtained by Eq. (5) are mainly distributed around the intercalated Pd atoms and exhibit an oscillatory behavior (Fig. 3c), which is related to the non-uniform charge density distribution caused by the bonding difference between the Pd and the top and bottom Se atoms. The different distribution between polarization and conduction electrons prevents the annihilation of Pout, where the conduction electrons distributed around the Zr atoms on the bottom layer cannot fully screen the vertical polarization originating from the intercalating Pd atom. Therefore, ferroelectricity and metallic behavior can coexist in T-PdZr2Se4, with a Pout of 3.10 pC/m. This mechanism is also responsible for the Pout of other sixteen non-magnetic FE compounds with metallic or half-metallic behavior, as summarized in Table 1, with values ranging from 0.43 to 9.61 pC/m, larger than the experimentally observed polarization of the metallic WTe2 bilayer (0.42 pC/m)36.

Fig. 3: The electronic band structures, projected density of states (PDOS) and electron density distribution of 2D T-PdZr2Se4 and T-CoTi2Te4.
figure 3

a, d Electronic band structures at the GGA +U level of T-PdZr2Se4 and T-CoTi2Te4, respectively. b, e PDOS for T- PdZr2Se4 and T- CoTi2Te4, where the top and bottom layers in (b) and (e) represent the ZrSe2 and TiTe2 in the top and bottom layers, respectively. c, f Polarization electron density ρ (z) (red line) and conduction electron density ρ (z) (blue line) distributions along the spatial z-direction of T-PdZr2Se4 and T-CoTi2Te4 monolayers.

Table 1 The band gap (eV) at GGA + U level, ground state (GS), spontaneous out-of-plane polarization (Pout) per unit cell in pC/m, FE transition barrier (EB) in meV/f.u. of 19 non-magnetic FE intercalation compounds

For the 40 selected FE materials, we further examined their magnetic ground states by comparing the total energy of FM with various types of AFM or FiM configurations. Among them, 21 systems exhibit both ferroelectricity and magnetism, including ten FM, nine AFM, and two FiM members (Table 2). It is worth noting that the presence of magnetism does not eliminate out-of-plane FE behavior in these intercalation compounds, even for metallic systems. For example, in T-CoTi2Te4, the contributions from the top and bottom TiTe2 layers to the conduction electrons are similar in the spin-up channel, while the conduction electrons in the spin-down channel originate from the top TiTe2 layer (Fig. 3e, f). In contrast, the polarization charge density mainly accumulates around the intercalated Co atoms, as displayed in Fig. 3f. The different origins of the two types of charges ensure the possibility of Pout in metallic 2D FE systems, no matter whether the system exhibits magnetism.

Table 2 The band gap (eV) at GGA + U level, ground state (GS), out-of-plane polarization (Pout) in pC/m, FE transition barrier (EB) in meV/f.u., magnetic atom (M-atom), average magnetic moment M (µB per magnetic atom), easy magnetization direction EMA, magnetic anisotropy energy MAE (meV per magnetic atom) and Curie temperature TC (K) or Néel temperature TN (K) simulated by MC for 21 multiferroic materials

ME coupling in multiferroic AM2X4 monolayers

In the following, we focus on the 21 intercalation materials with the coexistence of ferroelectricity and magnetism and perform a detailed analysis of their magnetism and ME coupling effects. For simplicity, we classify the 21 systems into three types according to the origin of magnetism: (i) type-a, in which the magnetism of the system originates from the magnetic atoms M in the top and bottom MX2 layers; (ii) type-b, in which the magnetism originates from the intercalated A atoms; (iii) type-c, in which both the M atoms in the MX2 layers and the intercalated A atoms provide magnetism simultaneously.

Type-a multiferroic materials, including T-PdCr2Te4, T-CuMn2Se4, T-AgMn2S4, T-AgMn2Se4, T-AgTc2S4, T-AgTc2Se4, T-AgCr2Te4, T-ZnCr2Se4, T-CdMn2Se4, T-CdCr2S4, T-CdCr2Se4, and T-CdCr2Te4, are obtained by inserting Pd/Cu/Ag/Zn/Cd atoms into T-CrX2, T-MnX2, or T-TcX2 bilayers. The intercalated A (A = Pd, Cu, Zn, Ag, and Cd) atoms with almost completely occupied 3d/4d orbitals do not contribute to magnetism, while the 3d orbitals of M (M = Cr, Mn, and Tc) atoms are partially filled, with more localized electrons and larger magnetic moments. Therefore, the magnetism comes from the M atoms in the MX2 bilayers, and the special bonding of the intercalated A atom results in negligible differences in the magnetic moments of the M atoms in the top and bottom layers (~0.1 µB).

The Heisenberg model was adopted to describe the magnetic interaction:

$$H=-\sum _{i,j}{J}_{i,j}{S}_{i}\cdot {S}_{j}-{A}_{z}\sum _{i}{\left({S}_{i}^{z}\right)}^{2}$$
(1)

where Si and Sj are the magnetic moments at i and j sites, respectively, Jij is the magnetic exchange coupling parameter between i and j sites, Az is the single-ion anisotropic energy, and the sum over i and j passes through all the magnetic ions. Considering the possible magnetic exchange interaction between the top and bottom MX2 layers, exchange interactions up to the fifth nearest neighbors are involved by comparing the energies of FM and different AFM configurations (Supplementary Fig. 7a and Supplementary Table 4). Here \({J}_{1}^{\text{intra}}\), \({J}_{2}^{\text{intra}}\), and \({J}_{3}^{\text{intra}}\) represent the intralayer coupling, while \({J}_{1}^{\text{inter}}\) and \({J}_{2}^{\text{inter}}\) are the interlayer interactions, as summarized in Supplementary Table 5. Taking T-CdCr2Te4 as a representative, \({J}_{1}^{\text{intra}}\) and \({J}_{3}^{\text{intra}}\) contribute most significantly to FM state, which can be explained by the Goodenough-Kanamori-Anderson (GKA) rules37, while the effect of \({J}_{2}^{\text{intra}}\) is too weak to affect the intralayer FM order. As for the interlayer magnetic interactions, both \({J}_{1}^{\text{inter}}\) and \({J}_{2}^{\text{inter}}\) favor FM order. Therefore, the Cr–Cr pairs in the intralayer and interlayer exhibit FM interactions. The magnetic anisotropy energy (MAE) is also an important parameter for characterizing the magnetic properties of a system, as a larger MAE can maintain long-range magnetic order against thermal fluctuation. It is defined as the energy difference between the magnetic configurations with in-plane and out-of-plane magnetization: MAE = Ein-plane − Eout-plane. Hence, a positive (negative) MAE represents an out-of-plane (in-plane) easy magnetization axis. The calculated MAE of T-CdCr2Te4 is −0.34 meV/Cr, indicating an in-plane magnetization orientation. Based on the magnetic coupling parameters and MAE, we performed MC simulations with the Heisenberg model to estimate the Curie temperature. As displayed in Supplementary Fig. 8a, T-CdCr2Te4 shows intrinsic FM order with a Curie temperature of 260 K, close to room temperature.

We performed AIMD simulations on T-CdCr2Te4 at different temperatures to examine the FE stability by analyzing the displacement of intercalated Cd atoms relative to the centrosymmetric position along the out-of-plane direction. As shown in Supplementary Fig. 8b, the average displacement of the Cd atoms (\(\bar{d}\)Cd) in z-direction follows a Gaussian distribution. At T = 250 K, \(\bar{d}\)Cd = 0.376 Å represents a macroscopic FE phase. When the temperature reaches T = 300 and 350 K, \(\bar{d}\)Cd rapidly decreases to 0.020 and 0.013 Å, respectively, and the peak position gradually moves toward zero and approaches the non-polar phase. This indicates that the FE transition temperature of 2D T-CdCr2Te4 is above room temperature and exhibits good multiferroic stability, as the FM transition temperature is close to room temperature (260 K). In addition, when T = 250 K, T-CdCr2Te4 remains in a stable structural configuration, and the intercalated atoms always remain at the tetrahedral site as shown in Supplementary Fig. 9a. At a temperature of 300 K, there is a change in the stacking mode due to thermodynamic disturbance, and a centrosymmetric non-polar state is formed (Supplementary Fig. 9b). This further confirms that the FE switching temperature of T-CdCr2Te4 is around room temperature and has good thermodynamic stability.

In addition, T-AgMn2S4, T-AgMn2Se4, T-ZnCr2Se4, T-CdCr2S4, T-CdCr2Se4, T-CdMn2Se4, and T-CuMn2Se4 also belong to the FM systems with Curie temperatures above room temperature, and the Curie temperature of T-AgMn2Se4 up to 525 K (Table 2 and Supplementary Fig. 10). In contrast, T-AgCr2Te4, T-PdCr2Te4, T-AgTc2S4, and T-AgTc2Se4 are found to be AFM systems, while the Néel temperature of T-AgTc2Se4 is 72 K. Excluding T-CdCr2S4 and T-AgTc2S4, other systems we consider feature in-plane easy magnetization axes, as detailed in Table 2. T-CdMn2Se4 exhibits the highest MAE of −0.99 meV/Mn among FM materials, T-AgTc2Se4 with AFM order reaching −2.49 meV/Tc, and T-AgTc2S4 with an in-plane easy magnetization axis shows an MAE as high as 2.09 meV/Tc.

Additionally, the breaking of time-reversal symmetry guarantees Heisenberg exchange interactions, while the strong spin-orbit coupling (SOC) induced by heavy Te element, along with the intrinsic inversion symmetry breaking from atoms, resulting in a significant DMI, naturally causing the formation of magnetic skyrmions. Therefore, we investigated the DMI of two equivalent FE phases and explored the possible topological magnetic texture and ME coupling behavior in T-CdCr2Te4. After considering DMI, the spin Hamiltonian can be rewritten as follows:

$${H}_{\text{spin}}=\mathop{\sum }\limits_{i < j}{J}_{\text{ij}}{\vec{{\boldsymbol{S}}}}_{{\bf{i}}}\cdot {\vec{{\boldsymbol{S}}}}_{{\bf{j}}}+{A}_{\text{z}}\mathop{\sum }\limits_{\text{i}}{\left({S}_{\text{i}}^{\text{z}}\right)}^{2}+\mathop{\sum }\limits_{i < j}{\vec{{\boldsymbol{D}}}}_{{\bf{ij}}}\cdot ({\vec{{\boldsymbol{S}}}}_{{\bf{i}}}\times {\vec{{\boldsymbol{S}}}}_{{\bf{j}}})$$
(2)

where the magnetic coupling parameter \({J}_{\text{ij}}\) can be extracted by the four-state energy mapping method38, and \({\vec{{\boldsymbol{D}}}}_{{\bf{ij}}}=[{D}_{\text{ij}}^{\text{x}},{D}_{\text{ij}}^{\text{y}},{D}_{\text{ij}}^{\text{z}}]\) is the DMI vector. The DMI components between the nearest Cr-Cr pairs of the top and bottom Cr layers have opposite directions (Fig. 4a, h). Since \({\vec{{\boldsymbol{D}}}}_{{\bf{ij}}}\) is a directional vector, we calculated \({J}_{\text{ij}}\) and \({\vec{{\boldsymbol{D}}}}_{{\bf{ij}}}\) along x, y, and xy directions for the first nearest neighbor Cr-Cr pairs of the top and bottom layers, respectively, as shown in Table 3 and Fig. 4. In the FE1 state, the D value for the top Cr layer is roughly triple that of the bottom layer, but this is inverted in the FE2 state, suggesting that DMI is very sensitive to structural asymmetry. Moreover, the D/|J| ratios of the top and bottom Cr layers in the FE1 state are about 9.03% and 20.66%, respectively, which shift to 20.80% and 9.52% in the FE2 state. Such a large DMI and D/|J| ratio may give rise to peculiar topological magnetic textures like magnetic skyrmions. At the same time, D/|J| values of the top and bottom Cr layers in the FE1 and FE2 states change with the switching of polarization. Therefore, one expects that FE polarization can regulate the topological magnetic behavior in different Cr layers.

Fig. 4: The top views of different spin textures from MC simulation snapshots at T = 0 K under different perpendicular magnetic fields.
figure 4

a, h Top views of top and bottom CrTe2 layers in 2D T-CdCr2Te4, respectively, while the red arrows illustrate the in-plane components of the DMI vectors between neighboring Cr –Cr pairs. According to the Moriya’s rule60, under C3v symmetry, the out-of-plane components of the DMI vector are arranged in a staggered pattern and can be ignored, while the in-plane components of the DMI vectors are perpendicular to the connecting lines between neighboring Cr atoms. bg represents the spin textures of top CrTe2 layer in FE1 states, while in represents the spin textures of bottom CrTe2 layer in FE2 states. The insets of (f) and (m) correspond to the enlarged image of anti-skyrmions observed of different Cr layers in FE1 and FE2 states, respectively.

Table 3 The magnetic coupling parameters (J, D, D/J) of nearest neighbor Cr–Cr pairs in 2D T-CdCr2Te4 along the x [(Cr–Cr)x], y [(Cr–Cr)y], and xy [(Cr–Cr)xy] directions of FE1 and FE2 states

To verify this speculation, we performed MC simulations for T-CdCr2Te4 in FE1 and FE2 states. The results are presented in Fig. 4, showing a wealth of topological magnetic excitations. As shown in Fig. 4b, the top Cr layer in the FE1 state displays a labyrinth-like spin texture (stripe domains) under low external fields (0 ~ 1.0 T), and the edge of the stripe domains wall becomes blurred with the increase of out-of-plane magnetic field strength. As the applied magnetic field reaches 2.4 T, the domain structure disintegrates, and anti-skyrmions appear (Fig. 4c), which can be maintained under a magnetic field of 2.4–3.17 T. Moreover, the diameter of anti-skyrmions decreases as the magnetic field increases, i.e., from 8.9 nm at 2.4 T to 3.8 nm at 3.17 T, which is much smaller than the reported diameters of 2D Cr(I, Cl)3 (10.5 nm at 0.8 T)39, Cr2Ge2Te6 (77 nm at 0.2 T)40, and the typical size of skyrmions in thin films (10–100 nm)41. Compared to domain walls in racetrack memory, the smaller size of skyrmions allows for faster data flow or higher data throughput with the same current density, resulting in higher information density. Therefore, T-CdCr2Te4 stands out as an excellent candidate for low-power consumption spintronic devices. As the magnetic field exceeds 3.5 T, the anti-skyrmions vanish, and the FM state becomes favorable (Fig. 4f). In other words, the spin structure of 2D T-CdCr2Te4 undergoes a phase transition with increasing magnetic field: stripe domain → skyrmion lattice → FM phase.

Unlike the top Cr layer, the lower Cr layer exhibits no topological magnetic properties in any magnetic field, which can be attributed to the difference in DMI between the top and bottom Cr layers. During the FE2 state, the top Cr layer behaves FM phase, while the bottom layer shows a phase transition akin to the top layer in the FE1 state, and the anti-skyrmions disappear as the magnetic field reaches 3.7 T, as illustrated in Fig. 4i-n. This indicates that the magnetic behavior of the top and bottom layers is reversed by polarization switching. Remarkably, the MX2 layer shows skyrmions with opposite chirality under different polarization directions (Fig. 4f, m), indicating that the FE polarization reversal can not only reversibly generate and annihilate magnetic skyrmions but also achieve chirality reversal of skyrmions, demonstrating interesting ME coupling phenomena.

So far, numerous theoretical and experimental studies have focused on controlling magnetic skyrmions via FE, most of which are based on heterostructures composed of FM and FE layers, where the topological magnetism of the FM layer is manipulated by polarization reversal of the FE layer42,43,44,45. To the best of our knowledge, there was a theoretical proposal of manipulating the magnetic skyrmions in the intercalation compound CoMo2S429, similar to the present study, where the chirality of skyrmions was tailored by FE polarization reversal in the intermediate Co layer. However, this differs from our strategy in which the modulation of topological magnetic textures occurs in the top and bottom CrTe2 layers. In other words, this work provides an approach to polarization-controlled topological magnetism.

We further investigated the effects of temperature and magnetic field on the topological spin textures of T-CdCr2Te4 monolayer. The magnetic behavior of T-CdCr2Te4 with temperature and magnetic field evolution is displayed in Fig. 5. At low temperatures (0 K ≤ T < 40 K), the system maintains a stripe domain phase within a small external magnetic field, evolving into a skyrmion lattice with the rise in magnetic field. It should be noted that the critical strengths of the magnetic field required for the formation and disappearance of skyrmions decrease as temperature increases. For example, anti-skyrmions emerge at B = 2.1 T and persist until B = 3.17 T at zero temperature, but at T = 25 K, they appear in an external magnetic field of 1.85 T and transform into FM phase when B exceeds 2.1 T. At higher temperatures, such as at T = 40 K, the system retains a stripe domain phase at lower magnetic field strengths, but it transforms directly into FM order instead of forming skyrmions when the magnetic field is further increased to 1.5 T. This implies that the critical temperature for the skyrmion lattice phase is below 40 K. As the temperature increased further (T > 60 K), the system exhibited a disordered phase due to strong thermal fluctuations. Therefore, the topological phase transition of 2D T-CdCr2Te4 can be achieved by modulating the external magnetic field and temperature, thereby generating and annihilating skyrmions. Furthermore, the top Cr layer in the FE2 state always maintains an FM state, while the magnetic behavior of the bottom Cr layer is similar to the top Cr layer in the FE1 state in terms of temperature and magnetic field effects, with the critical temperature of magnetic skyrmions of around 40 K (Supplementary Fig. 11).

Fig. 5: The spin textures views of the top CrTe2 layers in FE1 state under different temperatures and external perpendicular magnetic fields.
figure 5

The blue dashed line encircles the critical magnetic field when skyrmions disappear at different temperatures.

Among the considered systems, there are seven type-b multiferroic materials, including four T-AM2X4 (CoSc2S4, CoY2S4, CoZr2S4, CoZr2Se4) and three H-AM2X4 (CrMo2Se4, MnMo2S4, MnMo2Se4). In contrast to the type-a multiferroics, the intercalated species in the type-b multiferroics are transition metal atoms (Cr, Co, and Mn) with partially filled d orbitals, which are mainly responsible for the magnetism. The pristine MX2 (M = Sc, Y, Zr, Mo; X = S, Se) bilayers are non-magnetic, and the intercalation of A (A = Cr, Co, and Mn) atoms causes a transition from non-magnetic to magnetic state in the system.

The preferred magnetic order was determined by comparing the energy of different magnetic configurations (Supplementary Table 4), and three parameters for magnetic exchange interactions (J1, J2, and J3) are sufficient to describe the magnetic interactions as the magnetism is provided by the intercalated A atom within one layer. Take T-CoZr2S4 for example, its magnetic ground state is FM with the first nearest neighbor magnetic coupling coefficient J1 > 0. This can be explained by the GKA rules and is approximately 20 times larger than that of J2 and J3, dominating the magnetic interaction between the Co atoms. Meanwhile, the magnetization of T-CoZr2S4 prefers the out-of-plane direction with an MAE of 0.61 meV/Co, and the Curie temperature from MC simulation is about 70 K, as illustrated in Supplementary Fig. 10j. Simultaneously, T-CoZr2S4 remains in a FE state at 300 K without significant structural deformation (Supplementary Fig. 9c), indicating that it is thermodynamically stable at room temperature, and the FE transition temperature is higher than room temperature. Combined with the FM transition temperature, it is inferred that T-CoZr2S4 has room temperature stability and maintains multiferroic of coexisting ferroelectricity and ferromagnetism below 60 K. Moreover, T-CoZr2Se4 also has FM magnetic ground state, whereas T-CoSc2S4, T-CoY2S4, H-CrMo2Se4, H-MnMo2S4, and H-MnMo2Se4 favor AFM order, and T-CoY2S4 has the highest Néel temperature of 50 K (Table 2). Except for H-MnMo2Se4, the majority of type-b multiferroics prefer the out-of-plane direction as the easy magnetization axis, and T-CoY2S4 has a maximum MAE of 0.92 meV/Co.

The magnetic behavior (magnetic ground state and the direction of the easy magnetization axis) during the FE polarization reversal process was examined to investigate the ME coupling phenomena of T-CoZr2S4 (Fig. 6). T-CoZr2S4 maintains the ground state of FM during the initial phase of polarization reversal because the energies of FM are lower than those of AFM states. Nevertheless, the energies of AFM states decrease as get closer to the PE phase, turning the magnetic ground state into an AFM state. Subsequently, the magnetic ground state reverts to the FM state as the energy difference between the AFM and FM states increases. Interestingly, T-CoZr2S4 favors in-plane MAE with a greater value of −13.34 meV/Co in the PE phase, but in other states of the polarization reversal process, it adopts an out-of-plane easy magnetization axis. The magnetic ground state and MAE values for the other type-b multiferroics are also summarized in Supplementary Table 5. During the process of polarization reversal, T-CoY2S4 and H-MnMo2S4 exhibit a similar ME coupling phenomenon as T-CoZr2S4, while T-CoZr2Se4 and T-CoSc2S4 solely change the magnetic ground state, and H-MnMo2Se4 only shows changes in MAE (Supplementary Fig. 12).

Fig. 6: Changes in the magnetic ground state and MAE during the FE reversal process for T-CoZr2S4.
figure 6

The red, orange, and blue lines represent the energy differences of AFM1, AFM2, and AFM3 relative to the FM state, respectively, while the black line represents MAE for the most stable magnetic configurations.

In type-c multiferroics, pristine TiX2 (X = Se/Te) bilayers are non-magnetic semiconductors, and the intercalation of magnetic Co atoms produces two stable multiferroic materials with lower barriers, namely, T-CoTi2Se4 and T-CoTi2Te4. In addition to the intercalated Co atoms with intrinsic magnetic moment, induced magnetic moments also exist on the Ti atoms. The special coordination between the intercalated Co atoms and the Te atoms results in distinct magnetic moments between the top (Titop) and bottom (Tibot) layers of Ti. Therefore, one has to consider possible FiM configurations. The magnetic configurations and corresponding energies of FM, AFM, and FiM states for type-c multiferroics are presented in Supplementary Fig. 7c and Supplementary Table 6. Both T-CoTi2Se4 and T-CoTi2Te4 exhibit FiM ground states with total net magnetic moment of 0.24 µB/f.u. and 0.21 µB/f.u., respectively.

Using T-CoTi2Te4 as a representative, we have investigated the possible ME coupling phenomena in type-c multiferroic material. The band structures of FE1 and FE2 states exhibit metallic properties (Fig. 7a, b), but at the Fermi level, conduction electrons in the FE1 state are mainly contributed by spin-down electrons, while those in the FE2 state are predominantly dominated by spin-up channel, as illustrated in Fig. 7c, d. Furthermore, Titop and Tibot possess magnetic moments of 0.62 µB and 0.82 µB in the FE1 state, whereas these values become 0.82 µB and 0.62 µB in the FE2 state, indicating that reversing polarization direction may affect the spatial distribution of spin-polarized electrons. The more intuitive spin-polarized charge distribution diagrams can be found in Supplementary Fig. 13, which further demonstrates the influence of polarization on the spin-charge distribution. The CoTi2Te4 maintains a FiM order as its magnetic ground state in both FE1 and FE2 phases, but its easy magnetization axis changes from x-direction to z-direction after polarization reversal, further confirming the characteristics of ME coupling.

Fig. 7: The electronic band structure and PDOS of 2D T-CoTi2Te4.
figure 7

a, b Electronic band structures at GGA + U level of 2D T- CoTi2Te4 in FE1 and FE2 states, respectively. c, d Corresponding PDOS. The Fermi level is set to zero and marked by the black dotted line.

To summarize, we performed high-throughput first-principles calculations to filter potential 2D multiferroics induced by intercalated ions displacement among TMD intercalation compounds. By intercalating transition metal ions into the tetragonal-like vacancies of MX2 bilayers, we obtained 960 candidate 2D AM2X4 structures and identified 40 systems possessing high structural stability, large FE polarization, low FE switching barrier, and high transition temperature. We carefully discussed the intrinsic stabilization mechanism of polarization in four FE semiconductors and 31 FE metals/half-metals, and the out-plane polarization vary from 0.26 pC/m to 15.22 pC/m, comparable to those in typical 2D FE materials. Among the selected 40 FE intercalation compounds, there are 21 materials exhibit ferroelectricity and magnetism simultaneously. According to the origin of magnetism, we divided them into twelve type-a, seven type-b, and two type-c multiferroic materials, which exhibit different ME coupling behavior. In type-a multiferroic T-CdCr2Te4, the magnetism mainly arises from the CrTe2 bilayer, and skyrmion conversion between its top and bottom layers can be achieved through polarization switching. In type-b multiferroic T-CoZr2S4, the unfilled d orbitals of the intercalated Co atoms contribute to the magnetic moment. During the FE switching process, the magnetic ground state undergoes a transition from FM to AFM to FM, and the easy magnetization axis switches between out-of-plane and in-plane directions. The intercalated Co atoms can introduce magnetism into the non-magnetic TiSe2/TiTe2 bilayers, forming type-c multiferroics. In type-c multiferroic T-CoTi2Te4, both the spin-polarized charge spatial distribution and MAE change during polarization reversal. All these findings significantly expand the family of 2D ferroic materials and open a pathway for the development and implementation of nonvolatile logic and memory devices.

Methods

Electron density distribution

The conduction electron density \({\rho }_{c}(\vec{{\bf{r}}})\) is defined as the charge density within 0.05 eV near the Fermi level, and the conduction electron density distribution along z-direction was obtained by integrating \({\rho }_{c}(\vec{{\bf{r}}})\) within the xy plane:

$${\rho }_{c}({\rm{z}})=\iint {\rho }_{c}(\vec{{\bf{r}}})\,{dxdy}$$
(3)

The existence of center inversion symmetry in the paraelectric (PE) state ensures zero polarization, so the net polarization electron density \({\rho }_{P}(\vec{{\bf{r}}})\) is defined as the difference in charge density between the FE phase and the PE phase:

$${\rho }_{P}(\vec{{\bf{r}}})={\rho }_{{FE}}(\vec{{\bf{r}}})-{\rho }_{{PE}}(\vec{{\bf{r}}})$$
(4)

where \({\rho }_{{FE}}(\vec{{\bf{r}}})\) and \({\rho }_{{PE}}(\vec{{\bf{r}}})\) are the total electron density of the FE phase and the PE phase, respectively. Similar to the conduction electron density, we also integrate the polarization charges within the xy plane to obtain the distribution of the polarization electron density along the z-direction:

$${\rho }_{P}({\text{z}})=\iint [{\rho }_{{FE}}(\vec{{\bf{r}}})-{\rho }_{{PE}}(\vec{{\bf{r}}})]{dxd}y$$
(5)

Computational details

All calculations were performed using the density functional theory (DFT), as implemented in the Vienna ab initio simulation package (VASP)46. The ion-electron interactions were described using the projector-augmented plane wave (PAW) approach47, and the electron exchange-correlation interactions were described within the generalized gradient approximation parameterized by Perdew, Burke, and Ernzerhof (PBE)48. The planewave basis was expanded to 500 eV, with a convergence threshold of 107 eV in energy and 10−3 eV/Å in force, respectively. The Brillouin zone was sampled with the Γ-centered Monkhorst-Pack k-point grid49, with uniform spacing of 2π*0.02 Å−1. A vacuum space of more than 15 Å in z direction was added to avoid interactions between neighboring layers. The dynamic stability of all structures was assessed by their phonon dispersions, which were computed with the finite displacement method implemented in the Phonopy code50. The thermodynamic stability was evaluated by the formation energy, which is defined as Ef = (EAM2X4 − EA − 2 × EM − 4 × EX)/n, where EAM2X4 is the total energy of AM2X4 monolayers, EA, EM, and EX are the energies of the individual A, M, and X atoms in their most stable bulk phase, respectively, and n = 7 is the total number of atoms per AM2X4 formula. The FE energy transition barrier was evaluated using the climbing-image-nudged elastic band (CI-NEB) method51. The out-of-plane polarization was calculated considering the dipole correction52 and the in-plane polarization was estimated using the Berry phase method53. To account for the strong correlation effects involving localized d electrons, the effective Hubbard parameters Ueff = U − J were applied to all 3d and 4d transition metal elements, where U is the Coulomb repulsion, and J is the exchange parameter54,55,56. The Ueff values used in this work are summarized in Supplementary Table 1. The Curie or Néel temperature was evaluated using the Monte Carlo (MC) method57 with an 80 × 80 lattice. The FE switching temperatures were estimated by the ab initio molecular dynamics (AIMD) simulations within the NVT ensemble, which were performed on a 4 × 4 × 1 supercell for at least 10 ps with a time step of 1 fs58.

Micromagnetic simulations

As for the topological magnetism, we performed MC simulations for the CdCr2Te4 monolayer under different temperatures and magnetic fields to simulate the evolution of the spin textures using the Metropolis algorithm-based Spirit software59. The MC simulation considers periodic boundary conditions and operates in an 80 × 80 supercell containing 115,200 spin sites with a minimum MC step size of 6 × 105, and the initial magnetic moments were set to random configurations.